MONOTONIE D`UNE SUITE

f
un=n2f(x) = x2v0= 0,5
vn=f(vn−1)
un
n vn
sn= sin nπ
2
un+1 −un
un+1 −un= (n+1)2−n2= 2n+1 >0un
un=f(n)f[0; +∞[
unf
un=f(n)f:x7→ x2[0; +∞[
un
vnv0=1
2v1=1 4v2=1 16
vn+1 =f(vn)f[0; +∞[

un+1
un
>1un+1
un
<1
un6= 0 un
n6= 0 un=n2
un+1
un
=(n+ 1)2
n2=n+ 1 n2
=1 + 1
n2
>1
unun
wn=−2n
wn+1
wn
=−2n+1
−2n= 2 >1
wwn+1
wn
>1⇔wn+1 < wn
vn+1 =f(vn) = v2
n
v1< v0
n≥1vn< vn−1
vn+1 < vn
vn< vn−1f
[0; +∞[f(vn)< f(vn−1)vn+1 < vn
vn+1 < vnv
v1−v0<0
n≥1vn−vn−1<0
vn+1 −vn<0
vn+1 −vn=v2
n−v2
n−1= (vn−vn−1)(vn+vn−1)
vn−vn−1
vn+vn−1
v
vn+1 −vn<0

vn+1 −vn<0v
un
un=n−1
n+ 1
(u0= 5
un+1 =2un−1
2
un=en
n
un=2n+1
3n−2
(u0= 0
un+1 = 3 + un
3
un=n2−5n
un= 0,1n×n2
un=n+ cos n
u0= 5
un+1 =√un+ 2